The coast, the mountains, or the lakes, is exactly where I would stay in August, even though it's peak tourist season and prices will be higher. Way too hot to stay in a city like Florence or Rome.
You will need a rental car for anyplace outside of any city.
I would split my stay in weekly stays, to enjoy the various areas of the country.
For the coast of Tuscany I would stay in Castiglioncello/Quercianella (near Livorno) or Argentario Peninsula.
In the interior of Tuscany I would stay as high in altitude as I can. Consuma/Vallombrosa/Saltino is a good choice, Abetone area is another, Mount Amiata is another as well.
For the lakes you have several choices: Lake Como in the midlake, or lake Maggiore (Stresa?) or north lake Garda (Malcesine, Riva, Torbole, Limone).
For an alpine experience maybe the Dolomites mountains (Val Gardena?)
Any place in the Amalfi Coast would work. No need for a car here if you don't want.
Wherever you go, unless uphigh in the Alps (Dolomites) or the Appennino mountains mentioned above for Tuscany, make sure you have air conditioning or a body of water where to dive in on the hottest days.